In calculus, we have dealt with functions of x in two dimensional space. Multivariable Calculus, also known as Vector Calculus, deals with functions of two variables in 3 dimensional space, as well as computing with vectors instead of lines.
In single variable calculus, we see that y is a function of x
In multivariable calculus, z is a function of both x and y
Multivariable calculus extends the notion of differentiation and integration in three dimensional space. New concepts such as partial derivatives, directional derivatives, line integration and multiple integrals are introduced.
When plotting in 3 dimensional space, there needs to be new notation for various concepts that will be introduced. Multivariable calculus also introduces vectors, which may have been introduced in physics. Calculus dealing with vectors will take into consideration not only magnitude but direction as well.
Multivariable calculus is a broad field that applies to physics and other sciences more so than single variable calculus. Many of the theorems provided in vector calculus are essential for solving problems in physics, which are mostly multidimensional. Multivariable calculus studies the rates of change and works in more than two dimensions. In single variable calculus, an object might move through the air and we can calculate its velocity at any given point. In vector calculus, we can then take into account air resistance and the force of gravity acting on the object.
It is important to have a thorough understanding of the vectors and what they are used for before moving on to differentiation and integration in multivariable calculus. Since it is such a broad topic, we will start with an introduction to vectors in two and three dimensions as well as plotting lines and planes. Let's start with an introduction to vectors in two dimensional space.
Introduction to Vectors
Main Lesson: Vectors
An introduction to vectors in two dimensional space.
Key topics include Scalars and Vectors, Vector Notation, Vector Equalities and Operations, the Dot Product, the Angle between two vectors, and the Projection of a vector.
Vectors in Three Dimensional Space
Main Lesson: Vectors in 3 Dimensional Space
An introduction to working with vectors in three dimensional space and working with vectors in 3D.
Properties of Vectors
Main Lesson: Properties of Vectors
Various properties of vectors in 2D and 3D space.
Vector Functions of Lines and Planes
Main Lesson: Vector Functions
An introduction to vector functions and finding equations of lines and planes in two and three dimensional space.